论文标题
Banach空间中的代表定理：最小规范插值，正规学习和半分化反问题
Representer Theorems in Banach Spaces: Minimum Norm Interpolation, Regularized Learning and Semi-Discrete Inverse Problems
论文作者
论文摘要
从有限数量的采样数据点（测量）中构建或学习功能是科学和工程学中的基本问题。这通常是作为最小规范插值问题，正规学习问题或在某些功能空间中的半分化逆问题。适当空间的选择对于解决这些问题的解决方案至关重要。由重建功能（例如压缩感测和稀疏学习）的稀疏表示的动机，最近的许多研究兴趣都用于考虑某些Banach空间中的这些问题，以便获得其稀疏的解决方案，这是克服来自大多数实际应用的大数据本质的可行方法。本文的目的是对Banach空间中这些问题的代表定理进行系统研究。这些问题在Banach领域中有一些现有结果，所有这些问题都涉及隐式代表定理。我们旨在根据该定理获得明确的代表定理，然后将开发方便的解决方案方法。对于最小规范插值，显式代表定理使我们能够根据插值功能的线性组合的范围来表达imimum。为了开发有效的计算算法，我们建立了这些问题解决方案的定点方程式。我们透露，与希尔伯特空间不同，通常，在Banach空间中这些问题的解决方案可能无法减少到真正有限的维度问题（隐藏了某些无限的尺寸组件）。在特殊情况下，当Banach空间为$ \ ell_1（\ Mathbb {n}）$时，我们演示了如何删除该障碍的方式，将原始问题减少到真正有限的维度。
Constructing or learning a function from a finite number of sampled data points (measurements) is a fundamental problem in science and engineering. This is often formulated as a minimum norm interpolation problem, regularized learning problem or, in general, a semi-discrete inverse problem, in certain functional spaces. The choice of an appropriate space is crucial for solutions of these problems. Motivated by sparse representations of the reconstructed functions such as compressed sensing and sparse learning, much of the recent research interest has been directed to considering these problems in certain Banach spaces in order to obtain their sparse solutions, which is a feasible approach to overcome challenges coming from the big data nature of most practical applications. It is the goal of this paper to provide a systematic study of the representer theorems for these problems in Banach spaces. There are a few existing results for these problems in a Banach space, with all of them regarding implicit representer theorems. We aim at obtaining explicit representer theorems based on which convenient solution methods will then be developed. For the minimum norm interpolation, the explicit representer theorems enable us to express the infimum in terms of the norm of the linear combination of the interpolation functionals. For the purpose of developing efficient computational algorithms, we establish the fixed-point equation formulation of solutions of these problems. We reveal that unlike in a Hilbert space, in general, solutions of these problems in a Banach space may not be able to be reduced to truly finite dimensional problems (with certain infinite dimensional components hidden). We demonstrate how this obstacle can be removed, reducing the original problem to a truly finite dimensional one, in the special case when the Banach space is $\ell_1(\mathbb{N})$.
